Theorem zeqmznsub ≪ | index | src | ≫ 

theorem zeqmznsub (a b n: nat): $ modZ(n): a -ZN b = 0 <-> mod(n): a = b $;
StepHypRefExpression
1 bitr4 
(modZ(n): a -ZN b = 0 <-> b0 n |Z a -ZN b) -> (mod(n): a = b <-> b0 n |Z a -ZN b) -> (modZ(n): a -ZN b = 0 <-> mod(n): a = b)
2 zeqm03 
modZ(n): a -ZN b = 0 <-> b0 n |Z a -ZN b
3 1, 2 ax_mp 
(mod(n): a = b <-> b0 n |Z a -ZN b) -> (modZ(n): a -ZN b = 0 <-> mod(n): a = b)
4 bitr 
(mod(n): a = b <-> mod(n): b = a) -> (mod(n): b = a <-> b0 n |Z a -ZN b) -> (mod(n): a = b <-> b0 n |Z a -ZN b)
5 eqmcomb 
mod(n): a = b <-> mod(n): b = a
6 4, 5 ax_mp 
(mod(n): b = a <-> b0 n |Z a -ZN b) -> (mod(n): a = b <-> b0 n |Z a -ZN b)
7 eqmzdvdsub 
mod(n): b = a <-> b0 n |Z a -ZN b
8 6, 7 ax_mp 
mod(n): a = b <-> b0 n |Z a -ZN b
9 3, 8 ax_mp 
modZ(n): a -ZN b = 0 <-> mod(n): a = b

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)